On the discrete version of the Reissner-Nordstr\"om solution

TL;DR

This paper extends previous work on discrete Schwarzschild solutions to include charge, analyzing the discrete Reissner-Nordström solution within Regge calculus, and finds it avoids singularities while approximating the continuum solution at large scales.

Contribution

It introduces a discrete Reissner-Nordström solution in Regge calculus, incorporating electrodynamics and demonstrating the absence of singularities compared to the continuous case.

Findings

Discrete solution avoids singularities present in continuous solutions.

At large distances, the discrete solution closely approximates the continuum Reissner-Nordström metric.

Finite-difference form of the Hilbert-Einstein action is used for the Regge equations.

Abstract

This paper generalizes our previous paper on the discrete Schwarzschild type solution in the Regge calculus, the simplicial electrodynamics earlier considered in the literature is incorporated in the case of the presence of a charge. Validity of the path integral approach is assumed, of which the only consequence used here is a loose fixation of edge lengths around a finite nonzero scale (we have considered the latter earlier). In essence, the problem of determining the optimal background metric and electromagnetic field for the perturbative expansion generated by the functional integral is considered, for which the skeleton Regge and electrodynamic equations are analyzed. For the Regge equations, as we have earlier found, the Regge action on the simplest periodic simplicial structure and in the leading order over metric variations between 4-simplices can be substituted by a finite-difference form of the Hilbert-Einstein action (the piecewise constant metric there is defined by providing the vertices with coordinates). Thus we get the absence of the singularity inherent in the continuous solution. At the same time, the discrete solution is close to the continuum Reissner-Nordstr\"om one at large distances.